DISTRIBUTION OF THE PRIMES REPRESENTED BY ⌊ xn ⌋ IN SHORT INTERVALS

Guang-Liang Zhou; Ya-Fang Feng

doi:10.1216/rmj.2024.54.623

Abstract

Let $\left\lfloor x \right\rfloor$ be the largest integer not exceeding $x$ . For $0 < \theta < 1$ , let $\pi _\theta (x)$ denote the number of integers $n$ with $1 \le n \le x^\theta$ such that $\left\lfloor {x / n} \right\rfloor$ is prime. Recently, Ma, Chen and Wu obtained the interesting asymptotic formula

$\pi _\theta (x) = {{x^\theta} \over {(1 - \theta){\rm{log}}x}} + O(x^\theta ({\rm{log}}x)^{ - 2}),$

provided that ${{23} \over {47}} < \theta < 1$ . They further conjectured that this asymptotic formula can be extended to all $0 < \theta < 1$ . In this paper, we give an improvement of their result by showing that ${9 \over {19}} < \theta < 1$ is admissible.

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Guang-Liang Zhou. Ya-Fang Feng. "DISTRIBUTION OF THE PRIMES REPRESENTED BY $\left\lfloor {{x \over n}} \right\rfloor$ IN SHORT INTERVALS." Rocky Mountain J. Math. 54 (2) 623 - 629, April 2024. https://doi.org/10.1216/rmj.2024.54.623

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Received: 31 October 2022; Revised: 13 November 2022; Accepted: 3 January 2023; Published: April 2024

First available in Project Euclid: 7 May 2024

Digital Object Identifier: 10.1216/rmj.2024.54.623

Subjects:

Primary: 11N05

Secondary: 11A41

Keywords: Chebyshev estimate , distribution of the primes , exponential pair , the floor function

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